Efficient solvers for generalized incompressible flow problems with special emphasis on pressure Schur complement techniques for linearized Navier-Stokes equations and extensions

近年来，大量的工作致力于求解鞍点形式的大型（线性）系统问题。这种兴趣的原因是，这些问题出现在各种各样的技术和科学应用中。特别是，混合有限元方法在流体和固体力学等工程领域的日益普及已经成为这种鞍点系统的主要来源，因为它们通常来自不可压缩流动问题的离散化，例如由Navier-Stokes方程描述。由于鞍点系统的普遍性，在许多特殊应用和一般形式下，鞍点系统的离散化方面和数值解存在大量的文献。最近的一项综合调查[M]。Benzi， G.H. Golub， J. Liesen，鞍点问题的数值解，Acta Numerica 2005, pp.1-137]可以作为该主题的介绍，其中可以找到大量关于鞍点问题数值解的文献。这一调查表明，平稳和时变Stokes问题的情况已经或多或少地得到了解决，而线性化Navier-Stokes方程(包括对流部分（Oseen方程）和特别是非线性粘度（非牛顿方程）的有效求解器的发展。（如颗粒流），以及将Navier-Stokes方程与附加量耦合的扩展（k¿e湍流模型、Boussinesq方程、多相现象、粘弹性问题、流固相互作用）仍然是数值流动模拟领域的一项具有挑战性和重要的任务。在这个共同的项目中，我们将结合两个研究小组在不可压缩流体数值处理方面的理论和算法方面的专业知识，目的是开发、分析和实施改进的解决策略。特别地，我们将集中于非常数流问题，如。，非线性粘度小到中等雷诺数，因为它们通常出现在微型装置和毫微米反应器。主要的解决方法将基于压力舒尔补技术，该技术要么仅在压力空间中通过全局定义的近似预调节器构建，要么基于补丁智能定义的算子，包括完整流动方程的压力舒尔补，但在局部意义上。这些方法将应用于稳定和非协调Stokes单元的有限元离散所引起的鞍点问题，包括各种多项式空间。我们将从理论上分析开发的解决方法，并在FEM软件包中实现求解器组件，该组件可以直接验证和评估化学工程应用领域中广泛的原型流动配置。

In recent years, a large amount of work has been devoted to the problem of solving large (linear) systems in saddle point form. The reason for this interest is the fact that such problems arise in a wide variety of technical and scientific applications. In particular, the increasing popularity of mixed finite element methods in engineering fields such as fluid and solid mechanics has been a major source of such saddle point systems, as they typically arise from, the discretization of incompressible flow problems, for instance described by the Navier-Stokes equations. Because of the ubiquitous nature of saddle point systems, a wide literature exists on the discretization aspects and the numerical solution of such systems for many particular applications as well as in general form. A recent comprehensive survey [M. Benzi, G.H. Golub, J. Liesen, Numerical solution of saddle point problems, Acta Numerica 2005, pp.1-137] can serve as an introduction to the subject, where one can find enormous pointers to the literature on numerics for saddle point problems. This survey shows that the case of stationary and time-dependent Stokes problems has been more or less solved, while the development of efficient solvers for linearized Navier-Stokes equations including convective parts (Oseen equations) and particularly nonlinear viscosity (nonnewtonian, resp., granular flow), and moreover also for extensions which couple the Navier-Stokes equations with additional quantities (k ¿ e turbulence models, Boussinesq equations, multiphase phenomena, viscoelastic problems, fluid-structure interaction), is still a challenging and important task in the field of numerical flow simulation. In this common project, we will combine the special knowledge from each of both research groups, regarding theoretical as well as algorithmic aspects for the numerical treatment of incompressible fluids, with the aim to develop, to analyse and to implement improved solution strategies. In particular, we will concentrate on flow problems with non-constant, resp., nonlinear viscosity for small up to medium Re numbers as they typically arise in micro devices and milli-reactors. The main solution methodology will be based on pressure Schur complement techniques, which are either constructed via globally defined approximate preconditioners in the pressure space only, or which are based on patch wisely defined operators including the pressure Schur complement of the complete flow equations, but in a local sense. These approaches will be applied to saddle point problems arising from the FEM discretization with stable conforming as well as nonconforming Stokes elements, including various polynomial spaces. We will theoretically analyse the developed solution ethodology and realize the solver components in the FEM package FEATFLOW which directly allows a validation and evaluation for a wide class of prototypical flow configurations in the field of chemical engineering applications.